On the motive of the nested Quot scheme of points on a curve

Let C be a smooth curve over an algebraically closed field k, and let E be a locally free sheaf of rank r. We compute, for every d>0, the generating function of the motives [QuotC(E,n)]∈K0(Vark), varying n=(0≤n1≤⋯≤nd), where QuotC(E,n) is the nested Quot scheme of points, parametrising 0-dimensional subsequent quotients E↠Td↠⋯↠T1 of fixed length ni=χ(Ti). The resulting series, obtained by exploiting the Białynicki-Birula decomposition, factors into a product of shifted motivic zeta functions of C. In particular, it is a rational function.